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Time-Correlation Functions

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Time-Correlation Functions

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Charusita Chakravarty

Indian Institute of Technology Delhi

- Time Correlation Function: Definitions and Properties
- Linear Response Theory :
- Fluctuation-Dissipation Theorem
- Onsager’s Regression Hypothesis
- Response Functions

- Chemical Kinetics
- Transport Properties
- Self-diffusivity
- Ionic Conductivity
- Viscosity

- Absorption of Electromagnetic Radiation
- Space-time Correlation Functions

- Time-dependent trajectory of a classical system:
- Since the classical system is deterministic, a time-dependent
quantity can be written as :

- Correlation function as atime average over a trajectory:

- Time-correlation functions can be written as ensemble averages by summing over all possible initial conditions:
Probability of observing a microstate

- Limiting behaviour
- Alternative definition of time-correlation function in terms of deviations of time-dependent properties from mean values.

- Stationarity for systems with continuous interparticle forces, TCFs must be even functions of the time delay:
- Time-derivative with respect to time origins must be zero
- Short-time expansion of autocorrelation functions

Zero slope at origin

Rebound from

Solvent cage

D. Chandler

- Apply a weak perturbing field f to the system that couples to some physical property of the system
- Electric field/ionic motion
- Electromagnetic radiation/charges or molecular dipoles

System allowed to relax freely

Equilibrium

established

System prepared

in non-equilibrium

state by applying

perturbing field f

B(t)

D. Chandler

time=0

Let the time-dependent perturbation be such that

At t=0, the probability of observing a configuration:

How will the observed value of a quantity B(t) change with time when the perturbation is turned off at time t=0?

Time-dependent value of

B for a given set of initial

conditions

Integrate over initial

conditions of perturbed

system at t=0

Consider the effect of perturbations only upto first-order:

D. Chandler, Introduction to Modern Statistical Mechanics

For t>0, the observed value of B will be given by

Multiply the numerator and denominator by (1/Q) where Q is the partition function of the unperturbed system

Denominator

Numerator:

Time-dependent behaviour of B:

The relaxation of macroscopic non-equilibrium disturbances is governed by the same dynamics as the regression of spontaneous microscopic fluctuations in the equilibrium system

Macroscopic

relaxation

Equilibrium

Time-correlation

function

For a weak perturbation , we can define a response function:

The response of the system to an impulsive perturbation:

To correspond to the linear response situation studied earlier:

- Provides an alternative route to evaluate the time-dependent response as an integral over a time-correlation function

- Flux=-transport coefficient X gradient
- Non-equilibrium MD: create a perturbation and watch the time-dependent response
- Equilibrium MD: measure the time-correlation function

Consider an external field that couples to the position of a tagged particle such that

The steady state velocity of the tagged particle will then be:

Can one identify the mobility, as defined below, with the macroscopic velocity:

Fick’s Law:

Flux of diffusing species= Diffusivity X Concentration gradient

Combining with the equation of continuity derived by imposing conservation of mass of tagged particles, gives :

Diffusion Equation

If the original concentration profile is a delta-function, then the concentration profile at a later time (t) will be a d-dimensional Gaussian:

Consider the second-moment of one-dimensional distribution:

Einstein relation

By writing the displacement as

one can show that

This definition of self-diffusivity will be the same as that of the mobility derived from linear reponse theory

Consider an external electric field Ex applied to an ionic melt. Under steady state conditions, the system will develop a net current:

The ionic conductivity per unit volume, s, will be defined by:

Effect of the external field on the Hamiltonian:

When the field is switched off at t=0, the current will decay towards the zero value characteristic of the unperturbed system.

Rate of change of net

dipole moment

Charge on

particle i

velocity

particle i

To apply the relation:

to compute the time-dependent decay of the current, we set:

to obtain:

Conductivity per unit volume will then be given by:

Ionic Conductivity

Viscosity

- Let f(t) be a periodic, monochromatic disturbance:
- Time-dependent energy:
- Rate of absorption of energy:

- Time-dependent value of A reflects response to applied field:
- The average rate of absorption or energy dissipation is given by:
- For a periodic field:

- Fourier transform of response function is defined as:
- Compute average rate of absorption of energy over one time period T=2p/w

- Using linear response theory
- Absorption spectrum

- Let the quantity A coupled to the periodic perturbation obey SHO dynamics:
- Time-dependence of A:
- Absorption spectrum

- X-component of total dipole moment will couple to oscillating electric field .
- Independent dipole approximation:
- Perturbed Hamiltonian:
- Change in dipole moment with time:
- Thermal distribution of angular velocities, P(w), will be reflected in absorption profile

Number density at a point r at a time t:

Conservation of particle number:

Van Hove Correlation Function for a homogeneous fluid:

Can divide the double summation into two parts:

Self contribution

Distinct contribution

Fourier transform of the number density

Intermediate scattering function

Static structure factor

Dynamic structure factor

Sum rule

- D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications
- D. C. Rapaport, The Art of Molecular Dynamics Simulation (Details of how to implement algorithms for molecular systems)
- M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)
- Haile, Molecular Dynamics Simulation: Elementary Methods
- D. Chandler, Introduction to Modern Statistical Mechanics (Linear Response Theory)
- D. A. McQuarrie, Statistical Mechanics (Spectroscopic Properties)
- J.-P. Hansen and I. R. McDonald, The Theory of Simple Liquids (Almost everything)